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Derivatives of Sine

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Introduction

Garner

Mark Andrew Garner

Judy Land

Math Standard Level

April 30, 2008

Derivatives of Sine Functions

Method

1.  The following information was my investigation to find the derivative of the function .
1. The following is a graph for the function for . TI – 83 Graph Window Xmin = - Xmax = Xscl = Ymin = -5 Ymax = 5 Yscl = 1

1. The table below describes the behavior of the gradient of for .
 Gradient behavior in given points of x Y Gradient (+, , or 0) 0 +  1 0  0   1 0 0 0 + 1 0 0   1 0 0 +
1. Because the derivative of a function is the gradient at a given point, my conjecture for Middle 0 1 Relationship of dy/dx of f(x)=sinx versus Y-value of f(x)=cosx at a given X-value X-Value dy/dx f(x)=sinx f(x)=cosx 1 1  0 0   1 1 0 0 0 1 1 0 0  1 1 0 0 1 1 TI – 83 Graph Window Xmin = - Xmax = Xscl = Ymin = -5 Ymax = 5 Yscl = 1

The tables on the preceding page show that the derivative of at the following plotted points in increments of starting left to right x = . In the following graph, the points are connected to form , .

 TI – 83 Graph Window Xmin = - Xmax = Xscl = Ymin = -5 Ymax = 5 Yscl = 1  1. The following is my investigation of the derivatives of functions in the form . The red dots in the following graphs below represent the derivative of the function at a particular X-value.  Concluded from the graphs above, my conjecture for the derivative of the function , Conclusion can be written as . Consider the following graphs where the red line represents , , the gray line represents , and the red points represent the derivative of at an X-value.
 TI – 83 Graph Window Xmin = - Xmax = Xscl = Ymin = -5 Ymax = 5 Yscl = 1 Because of the chain rule: and ,

Thus: or =  In conclusion, from my investigation of the Derivative of sine functions, I have discovered the following:

• • , where the derivative is affected by
1. a the amplitude
2. b the frequency
3. c the phase shift
• The derivative of can be written as .

*In this investigation, there are limitations to graphs given as examples. The values substituted only occurred and, therefore, no negative values can be accounted for in the conjectures. Also, the graphs used only fall under and , and, thus, cannot confirm the conjecture outside of these limits.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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